You're missing the fact that a bite eventually kills you. It It gives you an extremely high fever that you inevitably succumb to. After that you turn. So the bite isn't what does it exactly but it speeds up the process by killing you.

The bite doesn't give you a fever. The infected wound gives you a fever.

Remember in the world of "The Walking Dead" everyone is infected with the disease. I've always thought that the title is more a taunt at the fact that everyone who is alive is actually still a zombie technically. Not just that the dead come back to life.

Also judging from the size of the wound on Lee's arm... Omid is still more of a prime candidate for zombification than Lee. Even with antibiotics Omid's wound is much worse both in placement and severity. Omid has already reopened his wound by simply walking. While Lee's bite is in a spot where less stress will be applied in fight or flight situations.